{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-0.0293466490732\n",
      "996.397409541\n"
     ]
    }
   ],
   "source": [
    "def neurous(sqrt = False):\n",
    "    num = 1000\n",
    "    x = np.ones(num)\n",
    "    if not sqrt:\n",
    "        w = np.random.randn(num)\n",
    "    else:\n",
    "        w = np.random.randn(num)/np.sqrt(num)\n",
    "    b = 0\n",
    "    z = np.dot(w, x) + b\n",
    "    return z\n",
    "\n",
    "zl = []\n",
    "for i in range(200000):\n",
    "    zl.append(neurous())\n",
    "\n",
    "print(np.mean(zl))\n",
    "print(np.var(zl))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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av0bSVGAOcDpwCtnDF98QEQeBZcAlwANkwWMWsLoBdTMzs36oNWz1pYi4WtIdwCH/04+I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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10cac6be0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n, bins, patches = plt.hist(zl, 100, alpha=0.8)\n",
    "plt.xlabel('z')\n",
    "plt.ylabel('Probability')\n",
    "plt.title('Histogram of $z=wx+b$')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c6139e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sigmod = lambda x:1./1.+np.exp(-x)\n",
    "al = sigmod(np.array(zl))\n",
    "\n",
    "n, bins, patches = plt.hist(al, 100, normed=1, alpha=0.8)\n",
    "plt.xlabel('a')\n",
    "plt.ylabel('Probability')\n",
    "plt.title('Histogram of $a=sigmod(z)$')\n",
    "# plt.axis([-1, 1, ])\n",
    "# plt.axis([40, 160, 0, 0.03])\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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NAGYCpwGnkE2++O6I2AcsBS4DHiJLHtOA1XWIzcxq5AcGLa/XiRElfTN93ilpVdelr5NL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\nTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10cabb390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "zl = []\n",
    "for i in range(200000):\n",
    "    zl.append(neurous(sqrt=True))\n",
    "    \n",
    "n, bins, patches = plt.hist(zl, 100, alpha=0.8)\n",
    "plt.xlabel('z')\n",
    "plt.ylabel('Probability')\n",
    "plt.title('Histogram of $z=wx+b$')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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uhnrTlipEkjS4pno11LbA0cATgW3H5lfVhI9BlSTNL1M9wf1JYHeaJ+d9k+apd57glqQH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\nTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10d011198>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "al = sigmod(np.array(zl))\n",
    "\n",
    "n, bins, patches = plt.hist(al, 100, normed=1, alpha=0.8)\n",
    "plt.xlabel('a')\n",
    "plt.ylabel('Probability')\n",
    "plt.title('Histogram of $a=sigmod(z)$')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}
